| Summary: | We show that a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><msub><mi>L</mi><mo>∞</mo></msub></mrow></semantics></math></inline-formula>-algebra <i>V</i> can be described by a nilpotent coderivation of degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> on coalgebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>P</mi><mo>*</mo></msup><mi>V</mi></mrow></semantics></math></inline-formula>. Based on this result, we can generalise the result of Lada to show that every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebra carries a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><msub><mi>L</mi><mo>∞</mo></msub></mrow></semantics></math></inline-formula>-algebra structure and every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><msub><mi>L</mi><mo>∞</mo></msub></mrow></semantics></math></inline-formula>-algebra carries an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebra structure. In particular, we obtain a pre-Lie <i>n</i>-algebra structure on an arbitrary partially associative <i>n</i>-algebra and deduce that pre-Lie <i>n</i>-algebras are <i>n</i>-Lie admissible.
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